How do you prove that the order of a in Zn (where n is an integer greater than or equal to 1) viewed as a group of order n with respect to addition is
Let thus, for .
We need to find the smallest such as,
By the properties of cyclic groups and the fact that is a generator it is equivalent to saying divides .
Thus, we need the smallest such that,
is an integer.
We can write it as, (by dividing by
The smallest such is of course,
Thus, the order of any element is,